# Joint Optimization of Preventive Maintenance, Spare Parts Inventory and Transportation Options for Systems of Geographically Distributed Assets

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Problem Statement

- A central warehouse is the primary source for all new spare parts and plays two roles in the spare part inventory flow—replenishing spare parts for the maintenance centers following a (s, S) replenishment policy [25], or providing spare parts directly to the assets as emergency orders when the maintenance order could not be satisfied from a maintenance center. Infinite inventory levels of spare parts are assumed for the central warehouse.
- A maintenance center fulfills maintenance orders from the nearby assets by shipping new undegraded spare parts to their operating sites. It is assumed to have finite inventory levels of spare parts and any maintenance order that cannot be immediately fulfilled by the maintenance center is serviced via an emergency order to the central warehouse.
- The term asset is used to refer to a machine that can be operated independently to generate revenue. It is assumed that there is a fleet of geographically dispersed assets in the system, labeled ${A}_{1},{A}_{2},\dots ,{A}_{J}$. An asset consists of multiple independent working parts and can only operate properly if all its parts behave properly.
- The term working part is used to refer to a basic unit of an asset. An asset ${A}_{j}$ ($1\le j\le J$) is assumed to be made up of ${K}_{j}$ serially connected parts, labeled ${P}_{j,1},{P}_{j,2},\dots ,{P}_{j,{K}_{j}}$. Degradation process of a part ${P}_{j,k}$ is characterized by a reliability function, ${\mathcal{D}}_{j,k}(\xb7)$, representing the distribution of that part’s usage time to failure. From the point of view of logistics, a working part on an asset corresponds to a certain type of a spare part that needs to be stored in the maintenance center. During a preventive or reactive maintenance intervention, a new spare part should be shipped either from the maintenance center or directly from the central warehouse to replace the degraded working part.

**Transportation**consists of shipping the ordered spare part to the asset ${A}_{j}$ from the maintenance center as a normal order, or from the central warehouse as an emergency order, with the lead times following the distributions $\mathcal{M}{\mathcal{T}}_{j}(\xb7)$ and $\mathcal{C}{\mathcal{T}}_{j}(\xb7)$, respectively.

- Lead time from the maintenance center to ${A}_{j}$ following the distribution $\mathcal{M}{\mathcal{T}}_{j}\left(\left(1+{u}_{j}\right)\ast t\right)$.
- Lead time from the central warehouse to ${A}_{j}$ following the distribution $\mathcal{C}{\mathcal{T}}_{j}\left(\left(1+{u}_{j}\right)\ast t\right)$.
- Expedited shipping cost to accelerate an RM delivery to the asset ${A}_{j}$ given by ${T}_{j}\ast {u}_{j}$.

**Execution**is essentially the process in which the target part on the asset is replaced with the newly delivered spare part, resulting in a maintenance intervention. The times needed to execute maintenance interventions will be referred to as repair times.

- Usage to failure of the part ${P}_{j,k}$ after PM following the distribution ${\mathcal{D}}_{j,k}\left(\frac{t}{\left(1-\alpha \right){v}_{j}+\alpha}\right)$.
- PM cost per order on the part ${P}_{j,k}$ given by ${M}_{j,k}\left({v}_{j}\right)={M}_{j,k}^{fix}+{M}_{j,k}^{add}\ast {v}_{j}$.
- PM repair time on the part ${P}_{j,k}$ given by $R{T}_{j,k}\left({v}_{j}\right)=R{T}_{j,k}^{fix}+R{T}_{j,k}^{add}\ast {v}_{j}$.

#### 2.2. Stochastic Optimization Formulation

## 3. Simulation-Based Optimization Approach

- Selection operator: A pair of parent solutions, namely $\left({X}^{\alpha},{Y}^{\alpha},{Z}^{\alpha},{U}^{\alpha},{V}^{\alpha}\right)$ and $\left({X}^{\beta},{Y}^{\beta},{Z}^{\beta},{U}^{\beta},{V}^{\beta}\right)$, are chosen from the current generation $g$ to mate and produce offspring candidate solutions for the next generation $g+1$, with a probability of selection being proportional to their fitness (in the GA literature, this is also known as fitness proportionate selection) [26].
- Crossover operator: For a pair of selected parent solutions, a single-point crossover operator is executed at a random point in each of the five chromosome portions, leading to five pairs of recombined chromosome portions, namely, $\left\{{X}^{a},{X}^{b}\right\}$, $\left\{{Y}^{a},{Y}^{b}\right\}$, $\left\{{Z}^{a},{Z}^{b}\right\}$, and $\left\{{V}^{a},{V}^{b}\right\}$. Then, an offspring solution is generated via randomly selecting a chromosome portion from each of the five pairs, while the remaining chromosome portions forms another offspring solution. The above-described crossover operator is pictorially illustrated in Figure 2.
- Mutation operator: To promote genetic diversity in the offspring population, each gene in an offspring solution chromosome is selected with a small probability (commonly referred to as the mutation probability), and its value is perturbed to an adjacent candidate in its candidate value set. (For example, assume that the PM triggering usage level ${x}_{j,k}$ takes values in ${X}_{j,k}=\left\{35,40,45,50\right\}$ and the current value for this gene is ${x}_{j,k}=40$. If the mutation operator is performed on this gene, the decision will mutate into either ${x}_{j,k}=35\mathrm{or}45$, with a small mutation probability.)

## 4. Results

#### 4.1. Baseline System and Restricted Systems

- (1)
- ${\mathbb{I}}_{1}=0$ denotes the existence of multiple PM operations with different quality levels, while ${\mathbb{I}}_{1}=1$ corresponds to the situation with perfect PM only. Thus, ${\mathbb{I}}_{1}=1$ implies fixing ${v}_{j}=1$ ($1\le j\le J$) in the formulation (1).
- (2)
- ${\mathbb{I}}_{2}=0$ denotes the existence of multiple spare parts shipping options for RM, while ${\mathbb{I}}_{2}=1$ corresponds to normal RM delivery only. Thus, ${\mathbb{I}}_{2}=1$ implies fixing ${u}_{j}=0$ ($1\le j\le J$) in the Formulation (1).
- (3)
- ${\mathbb{I}}_{3}=0$ denotes an (s, S) replenishment policy for spare parts inventory management in the maintenance center, while ${\mathbb{I}}_{3}=1$ indicates an (S − 1, S) replenishment policy in which only one spare part is shipped as a replenishment order. Thus, ${\mathbb{I}}_{3}=1$ implies fixing ${z}_{i}=1$ ($1\le i\le I$) in the Formulation (1).

#### 4.2. Sensitivity Analysis for Operating Costs under Integrated Policy

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Symbol | Description | Value |
---|---|---|

$J$ | Number of assets | 20 |

$CT$ | Lead time of replenishment delivery | 3 time units |

${H}_{i}$ | Inventory holding cost per unit time | 10 monetary unit/unit time |

${S}_{i}^{fix}$ | Fixed replenishment handling cost per order | 120 monetary unit/order |

${S}_{i}^{add}$ | Additional cost to have one more spare part added to replenishment order | 0 monetary unit/part |

${R}_{j,k}$ | RM cost per order | 1000 monetary unit/order |

$R{T}_{j,k}^{RM}$ | RM repair time per RM order | 0.5 time unit/order |

${M}_{j,k}^{fix}$ | Fixed PM cost per order | 200 monetary unit/order |

${M}_{j,k}^{add}$ | Additional PM cost to improve PM quality | 800 monetary unit/order |

$R{T}_{j,k}^{fix}$ | Fixed PM repair time per order | 0.4 time unit/order |

$R{T}_{j,k}^{add}$ | Additional repair time to improve PM quality | 0.1 time unit |

${E}_{j,k}$ | Additional charge of an emergency RM | 0 monetary unit/order |

${T}_{j}$ | Additional charge to accelerate RM delivery | 500 monetary unit/order |

- $\mathcal{M}{\mathcal{T}}_{j}(\xb7)$: Lead time distribution for the asset ${A}_{j}$ to obtain new spare parts from the maintenance center.
- $\mathcal{C}{\mathcal{T}}_{j}(\xb7)$: Lead time distribution for the asset ${A}_{j}$ to obtain new spare parts from the central warehouse.
- ${L}_{j}$: Penalty per unit downtime of the asset ${A}_{j}$.
- ${K}_{j}$: Number of working parts inside the asset ${A}_{j}$.

Asset | $\mathit{M}{\mathcal{T}}_{\mathit{j}}(\xb7)$ | $\mathcal{C}{\mathcal{T}}_{\mathit{j}}(\xb7)$ | ${\mathit{L}}_{\mathit{j}}$ (Monetary Unit/Unit Time) | ${\mathit{K}}_{\mathit{j}}$ | Corresponding Spare Part for Working Part |
---|---|---|---|---|---|

${A}_{1}$ | Weibull(1.1, 5) | Constant(3) | 400 | 4 | ${P}_{1,1}=S{P}_{1},{P}_{1,2}=S{P}_{2},{P}_{1,3}=S{P}_{3},{P}_{1,4}=S{P}_{4}$ |

${A}_{2}$ | Weibull(1.1, 5) | Constant(3) | 400 | 3 | ${P}_{2,1}=S{P}_{1},{P}_{2,2}=S{P}_{2},{P}_{2,3}=S{P}_{4}$ |

${A}_{3}$ | Weibull(1.1, 5) | Constant(3) | 400 | 2 | ${P}_{3,1}=S{P}_{1},{P}_{3,2}=S{P}_{3}$ |

${A}_{4}$ | Weibull(1.1, 5) | Constant(3) | 400 | 2 | ${P}_{4,1}=S{P}_{1},{P}_{4,2}=S{P}_{4}$ |

${A}_{5}$ | Weibull(1.1, 5) | Constant(3) | 400 | 3 | ${P}_{5,1}=S{P}_{1},{P}_{5,2}=S{P}_{3},{P}_{5,3}=S{P}_{4}$ |

${A}_{6}$ | Weibull(1.1, 5) | Constant(3) | 400 | 3 | ${P}_{6,1}=S{P}_{1},{P}_{6,2}=S{P}_{4},{P}_{6,3}=S{P}_{5}$ |

${A}_{7}$ | Weibull(1.1, 5) | Constant(3) | 400 | 2 | ${P}_{7,1}=S{P}_{1},{P}_{7,2}=S{P}_{5}$ |

${A}_{8}$ | Weibull(1.1, 5) | Constant(3) | 400 | 3 | ${P}_{8,1}=S{P}_{1},{P}_{8,2}=S{P}_{2},{P}_{8,3}=S{P}_{3}$ |

${A}_{9}$ | Weibull(1.1, 5) | Constant(3) | 400 | 2 | ${P}_{9,1}=S{P}_{2},{P}_{9,2}=S{P}_{3}$ |

${A}_{10}$ | Weibull(1.1, 5) | Constant(3) | 400 | 2 | ${P}_{10,1}=S{P}_{2},{P}_{10,2}=S{P}_{5}$ |

${A}_{11}$ | Weibull(2.2, 5) | Constant(3) | 800 | 4 | ${P}_{11,1}=S{P}_{1},{P}_{11,2}=S{P}_{2},{P}_{11,3}=S{P}_{3},{P}_{11,4}=S{P}_{4}$ |

${A}_{12}$ | Weibull(2.2, 5) | Constant(3) | 800 | 3 | ${P}_{12,1}=S{P}_{1},{P}_{12,2}=S{P}_{2},{P}_{12,3}=S{P}_{4}$ |

${A}_{13}$ | Weibull(2.2, 5) | Constant(3) | 800 | 2 | ${P}_{13,1}=S{P}_{1},{P}_{13,2}=S{P}_{3}$ |

${A}_{14}$ | Weibull(2.2, 5) | Constant(3) | 800 | 2 | ${P}_{14,1}=S{P}_{1},{P}_{14,2}=S{P}_{4}$ |

${A}_{15}$ | Weibull(2.2, 5) | Constant(3) | 800 | 3 | ${P}_{15,1}=S{P}_{1},{P}_{15,2}=S{P}_{3},{P}_{15,3}=S{P}_{4}$ |

${A}_{16}$ | Weibull(2.2, 5) | Constant(3) | 800 | 3 | ${P}_{16,1}=S{P}_{1},{P}_{16,2}=S{P}_{4},{P}_{16,3}=S{P}_{5}$ |

${A}_{17}$ | Weibull(2.2, 5) | Constant(3) | 800 | 2 | ${P}_{17,1}=S{P}_{1},{P}_{17,2}=S{P}_{5}$ |

${A}_{18}$ | Weibull(2.2, 5) | Constant(3) | 800 | 3 | ${P}_{18,1}=S{P}_{1},{P}_{18,2}=S{P}_{2},{P}_{18,3}=S{P}_{3}$ |

${A}_{19}$ | Weibull(2.2, 5) | Constant(3) | 800 | 2 | ${P}_{19,1}=S{P}_{2},{P}_{19,2}=S{P}_{3}$ |

${A}_{20}$ | Weibull(2.2, 5) | Constant(3) | 800 | 2 | ${P}_{20,1}=S{P}_{2},{P}_{20,2}=S{P}_{5}$ |

Spare Part Type | Weibull Distributed Time to Failure, $\mathbf{Weibull}\left(\mathit{k},\mathit{\lambda}\right):\mathit{k}$ for Shape, $\mathit{\lambda}$ for Scale | Expected Time to Failure, $\mathbb{E}\left(\mathit{S}{\mathit{P}}_{\mathit{h}}\right)$ | Standard Deviation of Time to Failure, $\mathbf{S}\mathbf{D}\left(\mathit{S}{\mathit{P}}_{\mathit{h}}\right)$ |
---|---|---|---|

$S{P}_{1}$ | $\mathrm{Weibull}\left(3.0,80\right)$ | 69.88 | 26.00 |

$S{P}_{2}$ | $\mathrm{Weibull}\left(4.0,\text{}100\right)$ | 93.06 | 25.45 |

$S{P}_{3}$ | $\mathrm{Weibull}\left(3.5,\text{}65\right)$ | 59.04 | 18.53 |

$S{P}_{4}$ | $\mathrm{Weibull}\left(3.5,\text{}70\right)$ | 63.58 | 19.95 |

$S{P}_{5}$ | $\mathrm{Weibull}\left(2.7,\text{}65\right)$ | 54.76 | 23.13 |

Symbol | Description | Value Set |
---|---|---|

${X}_{j,k}$ | A discrete real-number set for PM trigger ${x}_{j,k}$ * | $\left\{\mathbb{E}\left(S{P}_{h}\right)+\beta \xb7\mathrm{SD}\left(S{P}_{h}\right)\right\}$ where $\beta \in \left\{-2.5,-2.0,-1.5,-1.0,-0.5,0,0.5,1.0,1.5,2.0,2.5\right\}$ |

${Y}_{i}$ | A discrete integer set for re-order level ${y}_{i}$ | $\left\{-1,0,1,2,\dots ,20\right\}$ |

${Z}_{i}$ | A discrete integer set for batch size ${z}_{i}$ | $\left\{1,2,3\right\}$ |

${U}_{j}$ | A discrete real-number set for RM expedition rate ${u}_{j}$ | $\left\{0,0.5,1\right\}$ |

${V}_{j}$ | A discrete real-number set for PM recovery rate ${v}_{j}$ | $\left\{0,0.5,1.0,1.5,2.0\right\}$ |

**Table A5.**Parameters of the discrete event simulations, GA-related parameters, as well as computational times for the baseline system.

Description | Value | |
---|---|---|

General parameters | Time horizon, $T$ | 1825 time units |

Replication number | 100 | |

Parameters for GA | Population size | 60 |

Maximum iteration number | 500 | |

Maximum unchanged iteration | 30 | |

Crossover rate | 0.6 | |

Mutation rate | 0.05 | |

GA runs | 5 | |

Computational time of baseline system | Each GA iteration | 12.9 s |

Entire algorithm | $\le $10 h |

**Table A6.**Performance statistics of the baseline system and the restricted systems in Section 4.1.

System Index | R0 | R1 | R2 | R3 | R4 | R5 | R6 | Baseline |
---|---|---|---|---|---|---|---|---|

I1: Indicator for multi-mode PM option | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |

I2: Indicator for RM expedition option | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |

I3: Indicator for flexible replenishment option | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |

System uptime (%) | 94.40 | 94.91 | 95.29 | 94.54 | 95.71 | 94.85 | 95.18 | 95.66 |

Cumulative inventory holding times | 12,493.95 | 15,910.15 | 15,911.44 | 16,226.56 | 16,036.76 | 17,208.50 | 16,258.20 | 16,288.49 |

Cumulative replenishment order | 1311.20 | 1389.00 | 1388.72 | 614.71 | 1347.79 | 564.41 | 612.83 | 611.75 |

Cumulative number of PM orders | 591.16 | 446.99 | 557.64 | 623.36 | 229.26 | 482.55 | 579.95 | 380.05 |

Cumulative number of RM orders | 879.23 | 1012.68 | 903.93 | 859.42 | 1178.64 | 992.93 | 891.93 | 1084.70 |

Cumulative number of emergency orders | 10.36 | 4.53 | 4.66 | 7.39 | 3.83 | 6.99 | 7.27 | 7.07 |

Unit-time fixed PM cost (monetary unit) | 64.78 | 48.99 | 61.11 | 68.31 | 25.12 | 52.88 | 63.56 | 41.65 |

Unit-time added PM cost (monetary unit) | 259.14 | 185.01 | 244.44 | 273.25 | 81.43 | 197.03 | 254.22 | 143.44 |

Unit-time RM cost (monetary unit) | 481.77 | 554.89 | 495.30 | 470.92 | 645.83 | 544.07 | 488.73 | 594.36 |

Unit-time inventory holding cost (monetary unit) | 68.46 | 87.18 | 87.19 | 88.91 | 87.87 | 94.29 | 89.09 | 89.25 |

Unit-time replenishment cost (monetary unit) | 86.22 | 91.33 | 91.31 | 40.42 | 88.62 | 37.11 | 40.30 | 40.22 |

Unit-time downtime penalty (monetary unit) | 673.98 | 643.59 | 546.06 | 659.11 | 510.60 | 645.00 | 556.99 | 516.64 |

Unit-time RM acceleration cost (monetary unit) | 0.00 | 0.00 | 84.14 | 0.00 | 138.32 | 0.00 | 77.62 | 109.50 |

Unit-time emergency RM cost (monetary unit) | 10.36 | 4.53 | 4.66 | 7.39 | 3.83 | 6.99 | 7.27 | 7.07 |

**Table A7.**Factors F1–F6 used for the design-of-experiment (DOE) study in Section 4.2.

Factor | Description | Low vs. High Level | Relevant System Parameters Need to Be Scaled |
---|---|---|---|

F1 | Geographical dispersion level | 1.0 vs. 5.0 | $CT$ and $\mathcal{C}{\mathcal{T}}_{j}(\xb7)$ for $1\le j\le 20$ |

F2 | Inventory holding cost per unit time | 0.2 vs. 5.0 | ${H}_{i}$ for $1\le i\le 5$ |

F3 | Replenishment cost per order | 1.0 vs. 5.0 | ${S}_{i}^{fix}$ and ${S}_{i}^{add}$ for $1\le i\le 5$ |

F4 | PM quality improvement cost per order | 0.2 vs. 5.0 | ${M}_{j,k}^{add}$ for $1\le j\le 20,1\le k\le {K}_{j}$ |

F5 | Penalty cost per unit downtime | 0.2 vs. 5.0 | ${L}_{j}$ for $1\le j\le 20$ |

F6 | RM acceleration cost per order | 0.2 vs. 5.0 | ${T}_{j}$ for $1\le j\le 20$ |

**Table A8.**Operating costs under different system settings, with “L” denoting low level and “H” denoting high level.

F1 | F2 | F3 | F4 | F5 | F6 | Cost | F1 | F2 | F3 | F4 | F5 | F6 | Cost | F1 | F2 | F3 | F4 | F5 | F6 | Cost | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | L | L | L | L | L | L | 709.1 | 23 | H | L | L | L | H | H | 3244.8 | 45 | L | H | L | H | H | L | 1570.5 |

2 | L | L | H | L | L | L | 822.9 | 24 | H | L | H | L | H | H | 3095.0 | 46 | L | H | H | H | H | L | 3060.4 |

3 | H | L | L | L | L | L | 718.0 | 25 | L | L | L | H | L | H | 3237.3 | 47 | H | H | L | H | H | L | 3274.2 |

4 | H | L | H | L | L | L | 815.2 | 26 | L | L | H | H | L | H | 954.7 | 48 | H | H | H | H | H | L | 3581.8 |

5 | L | L | L | L | H | L | 2339.5 | 27 | H | L | L | H | L | H | 985.0 | 49 | L | H | L | L | L | H | 3791.0 |

6 | L | L | H | L | H | L | 2439.8 | 28 | H | L | H | H | L | H | 941.2 | 50 | L | H | H | L | L | H | 795.7 |

7 | H | L | L | L | H | L | 2383.2 | 29 | L | L | L | H | H | H | 998.3 | 51 | H | H | L | L | L | H | 1242.3 |

8 | H | L | H | L | H | L | 2451.1 | 30 | L | L | H | H | H | H | 4104.7 | 52 | H | H | H | L | L | H | 874.3 |

9 | L | L | L | H | L | L | 916.7 | 31 | H | L | L | H | H | H | 4162.4 | 53 | L | H | L | L | H | H | 1368.6 |

10 | L | L | H | H | L | L | 979.9 | 32 | H | L | H | H | H | H | 4167.9 | 54 | L | H | H | L | H | H | 3637.7 |

11 | H | L | L | H | L | L | 932.9 | 33 | L | H | L | L | L | L | 4183.9 | 55 | H | H | L | L | H | H | 3989.6 |

12 | H | L | H | H | L | L | 993.2 | 34 | L | H | H | L | L | L | 798.1 | 56 | H | H | H | L | H | H | 4151.8 |

13 | L | L | L | H | H | L | 2422.6 | 35 | H | H | L | L | L | L | 1241.3 | 57 | L | H | L | H | L | H | 4541.4 |

14 | L | L | H | H | H | L | 2495.5 | 36 | H | H | H | L | L | L | 927.4 | 58 | L | H | H | H | L | H | 1006.8 |

15 | H | L | L | H | H | L | 2461.2 | 37 | L | H | L | L | H | L | 1360.7 | 59 | H | H | L | H | L | H | 1329.7 |

16 | H | L | H | H | H | L | 2527.5 | 38 | L | H | H | L | H | L | 3046.8 | 60 | H | H | H | H | L | H | 1387.9 |

17 | L | L | L | L | L | H | 789.9 | 39 | H | H | L | L | H | L | 3237.6 | 61 | L | H | L | H | H | H | 1615.5 |

18 | L | L | H | L | L | H | 837.3 | 40 | H | H | H | L | H | L | 3465.2 | 62 | L | H | H | H | H | H | 4705.4 |

19 | H | L | L | L | L | H | 725.6 | 41 | L | H | L | H | L | L | 3768.8 | 63 | H | H | L | H | H | H | 4895.1 |

20 | H | L | H | L | L | H | 843.7 | 42 | L | H | H | H | L | L | 1010.1 | 64 | H | H | H | H | H | H | 5329.6 |

21 | L | L | L | L | H | H | 3024.1 | 43 | H | H | L | H | L | L | 1329.5 | ||||||||

22 | L | L | H | L | H | H | 709.1 | 44 | H | H | H | H | L | L | 1353.4 |

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**Figure 1.**Spare part logistic network considered in this paper. Illustration of the three-level logistic network (

**left-hand side**) and illustration of spare part inventory flows (

**right-hand side**). PM—Preventive Maintenance; RM—Reactive Maintenance.

**Figure 5.**Comparison of unit time operating costs for the baseline and restricted systems (

**left-hand side**), as well as a result of analysis of variance (ANOVA) of the unit time operating costs, with significance levels for the effects of factors ${\mathbb{I}}_{1}$, ${\mathbb{I}}_{2}$, and ${\mathbb{I}}_{3}$ (

**right-hand side**).

**Figure 6.**ANOVA analysis of the unit time operating costs under the integrated policy, with significance levels for the main/interaction effects of F1–F6.

**Table 1.**Notation used in the optimization Formulation (1). PM—Preventive Maintenance; RM—Reactive Maintenance.

Category | Symbol | Description |
---|---|---|

General notation | $i,j,k$ | Indices for spare part type ($i$), asset ($j$), working part ($k$) |

$T$ | Planning horizon | |

Candidate value set for decision vairlabe | ${X}_{j,k}$ | A discrete real-number set for PM trigger ${x}_{j,k}$ with values in $\left(0,\infty \right)$ |

${Y}_{i}$ | A discrete integer set for re-order level ${y}_{i}$ with values in $\left[-1,\infty \right)$ | |

${Z}_{i}$ | A discrete integer set for batch size ${z}_{i}$ with values in $\left[1,\infty \right)$ | |

${U}_{j}$ | A discrete real-number set for RM expedition rate ${u}_{j}$ with values in $\left[0,\infty \right)$ | |

${V}_{j}$ | A discrete real-number set for PM recovery rate $v$ with values in $\left[0,1\right]$ | |

Inventory-related terms | ${H}_{i}$ | Inventory holding cost per unit time for the spare part $S{P}_{i}$ |

${S}_{i}\left({z}_{i}\right)$ | Replenishment cost per order for the spare part $S{P}_{i}$ at the batch size ${z}_{i}$ | |

${h}_{i}$ | Cumulative inventory holding time of the spare part $S{P}_{i}$ | |

${s}_{i}$ | Cumulative replenishment order of the spare part $S{P}_{i}$ | |

PM | ${M}_{j,k}\left({v}_{j}\right)$ | Unit PM cost to perform PM on the part ${P}_{j,k}$ with the given ${v}_{j}$ |

${m}_{j,k}$ | Cumulative number of PM orders for the part ${P}_{j,k}$ | |

Normal RM | ${R}_{j,k}$ | Unit RM cost to perform RM on the part ${P}_{j,k}$ |

${r}_{j,k}$ | Cumulative number of RM orders for the part ${P}_{j,k}$ | |

Emergency RM | ${E}_{j,k}$ | Additional charge of an emergency RM on the part ${P}_{j,k}$ |

${e}_{j,k}$ | Cumulative number of emergency RM orders for the part ${P}_{j,k}$ | |

Downtime penalty | ${L}_{j}$ | Penalty cost per unit downtime of the asset ${A}_{j}$ |

Downtime on an Asset | ${l}_{j}$ | Total downtime that was observed on the asset ${A}_{j}$ |

Expedited Shipping | ${T}_{j}$ | Expedited shipping cost per RM order to the asset ${A}_{j}$ |

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## Share and Cite

**MDPI and ACS Style**

Wang, K.; Djurdjanovic, D. Joint Optimization of Preventive Maintenance, Spare Parts Inventory and Transportation Options for Systems of Geographically Distributed Assets. *Machines* **2018**, *6*, 55.
https://doi.org/10.3390/machines6040055

**AMA Style**

Wang K, Djurdjanovic D. Joint Optimization of Preventive Maintenance, Spare Parts Inventory and Transportation Options for Systems of Geographically Distributed Assets. *Machines*. 2018; 6(4):55.
https://doi.org/10.3390/machines6040055

**Chicago/Turabian Style**

Wang, Keren, and Dragan Djurdjanovic. 2018. "Joint Optimization of Preventive Maintenance, Spare Parts Inventory and Transportation Options for Systems of Geographically Distributed Assets" *Machines* 6, no. 4: 55.
https://doi.org/10.3390/machines6040055